What is the value of $\dfrac{d}{dx}\left(x^{-3}\right)$ at $x=3$ ?
Answer: Let's first find the expression for $\dfrac{d}{dx}\left(x^{-3}\right)$ and then evaluate it at $x=3$. The derivative can be found using the power rule : $\dfrac{d}{dx}(x^n)=n\cdot x^{n-1}$ (Remember that this applies even when $n$ is a negative number.) $\begin{aligned} &\phantom{=}\dfrac{d}{dx}\left(x^{-3}\right) \\\\ &=-3x^{-3-1} \gray{\text{The power rule}} \\\\ &=-3x^{-4} \end{aligned}$ So we found that $\dfrac{d}{dx}\left(x^{-3}\right)=-3x^{-4}$, which can also be written as $-\dfrac{3}{x^4}$. Now let's plug ${x=3}$ : $\begin{aligned} -\dfrac{3}{({3})^4}&=-\dfrac{3}{81} \\\\ &=-\dfrac{1}{27} \end{aligned}$ In conclusion, the value of $\dfrac{d}{dx}\left(x^{-3}\right)$ at $x=3$ is $-\dfrac{1}{27}$.